Without a picture of the puzzle, we're likely to make unhelpful suggestions, however...
Suggestions to take it to a repairer are defeatist! OK, not all of us revel in this sort of mechanical puzzle, but really, no particular skill is required.
If you've got the buttons collated into rows, and in the right order along each row, is the problem simply(!) stuffing them through the holes? In other words, no sorting or identification required? One suggestion: stick the rods of each row to the cardboard strip you've already fashioned, with adhesive tape, so they form straight, correctly-spaced rows ('plates' if you like), then you only need three hands to wiggle the entire row around. Hopefully, the 'hole plate' can be tilted so you can fit one row at a time. BluTac or Plasicene (spelling?) are useful as temporary restraints.
Second suggestion is to dismantle the entire mechanism. It's not difficult, just tedious, and you need to invent your own mental-lapse-proof method of holding the removed components in their correct order. Label everything, take notes, photograph! Then re-build one rod at a time, starting with the counter-bass row (how can you do it any other way?). Hopefully, you'll find it easier than you think. I've worked on a mid-sized, 120 bass box, in which some of the counter-bass and bass buttons just wouldn't go in, without temporarily removing several 'pipes'. I don't think that's 'normal', but a side-effect of the small size. It was a fiddle, but patience won, in the end.
And for your next accordion, get one with a removable bass machine!